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Barbara Smith is interviewing candidates to be her secretary. As she interviews
the candidates, she can determine the relative rank of the candidates
but not the true rank. Thus, if there are six candidates and their true rank is
6, 1, 4, 2, 3, 5, (where 1 is best) then after she had interviewed the first three
candidates she would rank them 3, 1, 2. As she interviews each candidate,
she must either accept or reject the candidate. If she does not accept the
candidate after the interview, the candidate is lost to her. She wants to decide
on a strategy for deciding when to stop and accept a candidate that will
maximize the probability of getting the best candidate. Assume that there
are n candidates and they arrive in a random rank order.
(a) What is the probability that Barbara gets the best candidate if she interviews
all of the candidates? What is it if she chooses the first candidate?
(b) Assume that Barbara decides to interview the first half of the candidates
and then continue interviewing until getting a candidate better than any
candidate seen so far. Show that she has a better than 25 percent chance
of ending up with the best candidate.
I'm a little confused on this problem because there isn't a set number of candidates, but here's what I'm thinking so far
a) if she interviews all the candidates, then the probability of getting the best candidate is 1/n because she would reject the first (n-1) candidates and accept the last one
if she interviews only the first candidate, then the probability is also 1/n
I'm not really sure what to do for part b, though...